A computer network uses polynomials over \(GF(2)\) for error checking with 8…
2017
A computer network uses polynomials over \(GF(2)\) for error checking with 8 bits as information bits and uses \(x^{3}+x+1\) as the generator polynomial to generate the check bits. In this network, the message 01011011 is transmitted as:
- A. 01011011010
- B. 01011011011
- C. 01011011101
- D. 01011011100
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Correct answer: C
Key idea: compute the CRC remainder by dividing the message (with appended zeros) by the generator over GF(2).
Step 1: Append three zeros (degree of generator = 3) to the 8-bit message 01011011 → 01011011000.
Step 2: The generator polynomial x^3 + x + 1 corresponds to divisor bits 1011.
Step 3: Divide 01011011000 by 1011 using binary (XOR) division over GF(2). The division leaves a remainder of 101.
Step 4: Append the remainder 101 to the original 8-bit message: 01011011 + 101 = 01011011101.
Final transmitted message: 01011011101
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